The generalized infimal convolution can be used to naturally prove some dominated monotone additive extension theorems
Abstract
By using a particular case of the generalized infimal convolution, we provide an instructive proof for a particular case of a dominated monotone additive extension theorem of Benno Fuchssteiner.
Keywords
preordered groups; generalized infimal convolution; dominated monotone additive extensions
References
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24. Nef W., Monotone Linearformen auf teilgeordneten Vektorräumen, Monatsh. Math. 60 (1956), 190–197.
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27. Roth W., Hahn–Banach type theorems for locally convex cones, J. Austral. Math. Soc. (Ser. A) 68 (2000), 104–125.
28. Simons S., From Hahn–Banach to Monotonicity, Springer-Verlag, Berlin, 2008.
29. Strömberg T., The operation of infimal convolution, Dissertationes Math. 352 (1996), 1–58.
30. Száz Á., The intersection convolution of relations and the Hahn–Banach type theorems, Ann. Polon. Math. 69 (1998), 235–249.
31. Száz Á., A reduction theorem for a generalized infimal convolution, Tech. Rep., Inst. Math., Univ. Debrecen 11 (2009), 1–4.
32. Száz Á., The infimal convolution can be used to derive extension theorems from the sandwich ones, Acta Sci. Math. (Szeged) 76 (2010), 489–499.
33. Száz Á., The intersection convolution of relations, Creative Math. Inf. 19 (2010), 209– 217.
2. Anger B., Lembcke J., Hahn–Banach type theorems for hypolinear functionals on preordered topological vector spaces, Pacific J. Math. 54 (1974), 13–33.
3. Aumann G., Über die Erweiterung von additiven monotonen Funktionen auf regulär geordneten Halbgruppen, Arch. Math. 8 (1957), 422–427.
4. Badora R., On the Hahn–Banach theorem for groups, Arch. Math. 86 (2006), 517–528.
5. Boccuto A., Candeloro D., Sandwich theorems and applications to invariant measures, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 511–524.
6. Burai P., Száz Á., Homogeneity properties of subadditive functions, Ann. Math. Inform. 32 (2005), 189–201.
7. Buskes G., The Hahn–Banach Theorem surveyed, Dissertationes Math. 327 (1993), 1–49.
8. Day M.M., Normed Linear Spaces, Springer-Verlag, Berlin, 1962.
9. Figula Á., Száz Á., Graphical relationships between the infimum and intersection convolutions, Math. Pannon. 21 (2010), 23–35.
10. Fuchssteiner B., Sandwich theorems and lattice semigroups, J. Funct. Anal. 16 (1974), 1–14.
11. Fuchssteiner B., Lusky W., Convex Cones, North-Holland, New York, 1981.
12. Gajda Z., Sandwich theorems and amenable semigroups of transformations, Grazer Math. Ber. 316 (1992), 43–58.
13. Gajda Z., and Kominek Z., On separation theorems for subadditive and superadditive functionals, Studia Math. 100 (1991), 25–38.
14. Glavosits T., Kézi Cs., On the domain of oddness of an infimal convolution, Miskolc Math. Notes 12 (2011), 31–40.
15. Glavosits T., Száz Á., The infimal convolution can be used to easily prove the classical Hahn–Banach theorem, Rostock. Math. Kolloq. 65 (2010), 71–83.
16. Glavosits T., Száz Á., A Hahn–Banach type generalization of the Hyers–Ulam theorem, An. St. Univ. Ovidius Constanta 19 (2011), 139–144.
17. Glavosits T., Száz Á., Constructions and extensions of free and controlled additive relations, Tech. Rep., Inst. Math., Univ. Debrecen 1 (2010), 1–49.
18. Grilliot T.J., Extensions of algebra homomorphisms, Michigan Math. J. 14 (1967), 107–116.
19. Jameson G., Ordered Linear Spaces, Springer-Verlag, Berlin, 1970.
20. Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, Polish Sci. Publ. and Univ. Šlaski, Warszawa, 1985.
21. Moreau J.J., Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. 49 (1970), 109–154.
22. Namioka I., Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957), 1–50.
23. Narici L., Beckenstein E., The Hahn–Banach theorem: the life and times, Topology Appl. 77 (1997), 193–211.
24. Nef W., Monotone Linearformen auf teilgeordneten Vektorräumen, Monatsh. Math. 60 (1956), 190–197.
25. Peressini A.L., Ordered Topological Vector Spaces, Harper and Row Publishers, New York, 1967.
26. Plappert P., Sandwich theorem for monotone additive functions, Semigroup Forum 51 (1995), 347–355.
27. Roth W., Hahn–Banach type theorems for locally convex cones, J. Austral. Math. Soc. (Ser. A) 68 (2000), 104–125.
28. Simons S., From Hahn–Banach to Monotonicity, Springer-Verlag, Berlin, 2008.
29. Strömberg T., The operation of infimal convolution, Dissertationes Math. 352 (1996), 1–58.
30. Száz Á., The intersection convolution of relations and the Hahn–Banach type theorems, Ann. Polon. Math. 69 (1998), 235–249.
31. Száz Á., A reduction theorem for a generalized infimal convolution, Tech. Rep., Inst. Math., Univ. Debrecen 11 (2009), 1–4.
32. Száz Á., The infimal convolution can be used to derive extension theorems from the sandwich ones, Acta Sci. Math. (Szeged) 76 (2010), 489–499.
33. Száz Á., The intersection convolution of relations, Creative Math. Inf. 19 (2010), 209– 217.
GlavositsT., & Száz Árpád. (2011). The generalized infimal convolution can be used to naturally prove some dominated monotone additive extension theorems. Annales Mathematicae Silesianae, 25, 67-100. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14023
Tamás Glavosits
glavo@science.unideb.hu
Institute of Mathematics, University of Debrecen, Hungary Hungary
Institute of Mathematics, University of Debrecen, Hungary Hungary
Árpád Száz
Institute of Mathematics, University of Debrecen, Hungary Hungary
Institute of Mathematics, University of Debrecen, Hungary Hungary
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